Carols, today, mean “songs that are sung at Christmas.” During the Middle Ages, a carol meant a ring dance accompanied by a song.  These were often done at Christmas, but they were not exclusively Christmas fare. It’s tempting to say that anything which sounds archaic is probably an authentic medieval carol, but we don’t have good records of popular music from the Middle Ages (and many modern Christmas traditions are Victorian in origin). “The Holly and the Ivy” is probably one authentic medieval carol, as “Holly-versus-Ivy” games were popular medieval Christmas games  and the song features imagery consistent with medieval Christianity. With dancing and singing, much fun and revelry occurred on Christmas.
Gifts were given at Christmas during the Middle Ages, but not in the same way that they are given today. The king was expected to give gifts to his retainers, and members of the upper classes gave gifts to their servants. Norman Davis notes that gifts were recorded in records that survive to this day—for example, Chaucer and his wife often received silver or cloth.  I have not found instances of lower-ranking people giving gifts to higher-ranking people. If gifts were not given on Christmas itself, they were sometimes exchanged on New Year’s Day.
The Boar’s Head Feast
Information on the Boar’s Head Feast is sketchy, but this probably was a tradition observed in various forms throughout the Middle Ages. Some claim Germanic origins for it, as boars were reputed to be sacrificed to the Norse god Freyr at Yultide, and the boar’s head would be served at a banquet.  Boar hunting and the singing of the Boar’s Head Carol (here’s an authentic one!) were commonly observed around Christmastime and New Year’s Day during the Middle Ages.  Many examples of the boar-hunt survive from medieval literature. The most famous of them may be the one from Sir Gawain and the Green Knight—boars were usually hunted “from Christmas to Candlemas.” 
I don’t think that this is a medieval thing. I’ve never found a reference to yule logs in medieval literature. Some say that yule logs come from Germanic pagan practices that were continued in England until modern times, but I’ve yet to find definite research.
Arthurian myth almost always includes feasting, jousting, and tournaments as part of Christmas festivities.  Christmas was one of the few feast days in which King Arthur would wear his crown and hold a full court. He would also refuse to dine until an adventure had occurred or been related, a tradition that also makes it into some of the Robin Hood ballads. As to whether or not medieval people actually held tournaments on Christmas, if it was done, it was likely done in imitation of Arthurian myth. Many games were also part of the Christmas festivities. Men and women would often take opposing sides during such games. Sir Gawain and the Green Knight offers an example of such a game, the exact nature of which is unknown, but appears that a hood is given as a prize to the winner (medieval hoods were not attached to other clothing as they are today). 
I recognize that many of my readers do not celebrate Christmas. However, I do, and so I wish you all a wonderful Christmas!
- Oxford English Dictionary, 2nd ed., s.v. “carol,” http://www.oed.com/view/Entry/28123?isAdvanced=false&result=1&rskey=174sKQ& (accessed November 12, 2012).
- Davis, Norman, ed. “Sir Gawain and the Green Knight.” Medieval English Literature, ed. Thomas J. Garbáty. (1968, repr. Long Grove, Illinois: Waveland Press, 1984), 285.
- Ibid., 257.
- Spears, James E. “The ‘Boar’s Head Carol’ and Folk Tradition.” Folklore 85.3 (1974): 196.
- Davis, Norman, “Sir Gawain,” 305.
- Ibid., 256.
- Ibid., 285.